# How to Find the Period of a Function?

A periodic function is a function that repeats itself at regular intervals. In the following step-by-step guide, you will learn how to find the period of a function.

The time interval between two waves is known as a period, while a function that repeats its values at regular intervals or periods is known as a periodic function. In other words, the periodic function is a function that repeats its values after each particular period.

## Step by step guide to a periodic function

A function \(y = f (x)\) is a periodic function in which there exists a positive real number \(P\) such that \(f (x + P) = f (x)\), for all \(x\) belong to real numbers. The smallest value of a positive real number \(P\) is called the **fundamental period of a function**. This fundamental period of a function is also called the **function period** in which the function repeats itself.

\(\color{blue}{f(x+P)=f(x)}\)

**Note:** the sine function is a periodic function with a period of \(2π\). \(sin(2π + x) = sinx\).

Below are diagrams of some of the periodic functions. The graph of each of the following periodic functions has translational symmetry.

### Periods of some important periodic functions

The period of a function helps us to know the interval, after which the range of the periodic function is repeated. The domain of a periodic function \(f(x)\) includes the real number values of \(x\), and the range of a periodic function is a limited set of values within an interval. The length of this repeating interval, or the interval after which the range of the function repeats itself, is called the **period of the periodic function**.

The periods of some important periodic functions are as follows:

- The period of \(sinx\) and \(cosx\) is \(2π\).
- The period of \(tanx\) and \(cotx\) is \(π\).
- The period of \(secx\) and \(cosecx\) is \(2\).

### Properties of periodic functions

The following features are useful for a deeper understanding of the concepts of periodic function:

- The graph of a periodic function is symmetric and repeats itself along the horizontal axis.
- The domain of the periodic function includes all values of real numbers, and the range of the periodic function is defined for a fixed interval.
- The period of a periodic function against which the period is repeated is equal to the constant over the whole range of the function.
- If \(f (x)\) is a periodic function with period \(P\), \(\frac{1}{f(x)}\) will also be a periodic function with the same fundamental period \(P\).
- If \(f(x)\) is a periodic function with a period of \(P\), then \(f(ax + b)\) is also a periodic function with a period of \(\frac {P}{|a|}\).
- If \(f(x)\) is a periodic function with a period of \(P\), then \(af(x) + b\) is also a periodic function with a period of \(P\).

### Periodic Function – Example 1:

Find the period of the periodic function \(y=sin(4x + 5)\).

**Solution:**

The period of \(sinx\) is \(2π\), and the period of \(sin(4x + 5)\) is :

\(\frac{2π}{4}=\frac{π}{2}\)

Therefore, the period of \(sin(4x + 5)\) is \(\frac{π}{2}\).

### Periodic Function – Example 2:

Find the period of the periodic function \(y=9 cos(6x + 4)\).

The period of \(cosx\) is \(2π\), and the period of \(9 cos(6x + 4)\) is:

\(\frac{2π}{6}=\frac{π}{3}\)

Therefore, the period of \(9 cos(6x + 4)\) is \(\frac{π}{3}\).

## Exercises for Periodic Function

### Find the period of the function.

- \(\color{blue}{y= tan3x + sin\frac{5x}{2}}\)
- \(\color{blue}{y=sec(\pi x-2)}\)
- \(\color{blue}{y=cot(-(\frac{2\pi}{3})x)}\)
- \(\color{blue}{\:y=cos\left(-\left(\frac{2}{3}\right)x-\pi \right)}\)

- \(\color{blue}{4\pi}\)
- \(\color{blue}{2}\)
- \(\color{blue}{\frac{3}{2}}\)
- \(\color{blue}{3\pi}\)

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